I cannot imagine anything being simpler than that! The "intuition" about the logic is precisely the meaning of "inverse"- it has nothing to do with "columns" or "rows".
Hi guys,
I would like to intuitively understand why the following is true:
Let A be a square, invertible matrix. let C be A's left inverse, and B A's right inverse.
Then it follows: B=C
I am aware of the simple proof:
C=C(AB)=CAB=(CA)B=B
yet, this gives away no intuition about the logic behind this claim.
The left inverse is acting on A's columns, the right one is acting on A's rows.
This seems almost like magic.
How would you explain this in child's terms?
Thanks.
I'm going by the natural definition of left or right inverse. in square matrices they are the same and only then can you say "inverse" without specifying left or right.
Sure we can look at this in a high level manner, as a general algebraic structure satisfying associativity and existence of unit element. But then we'd be ignoring the beautiful properties of matrices.
AB means a series of linear transformations on columns of A, according to B.
if AB=I, one cannot immediately see why BA = I as well. BA meaning a series of linear transformations on rows of A according to B, which is, so to say, a completely different operation.
The simple proof above, simply does not answer my question. at least not directly.
Tell you what: why don't you give us a list of assumptions (axioms) that you both believe and understand, and that are relevant to the question at hand. List also any other theorems you buy into. Then state the theorem you don't understand, and then perhaps we could produce a constructive proof of the theorem that might enable you to understand.
Hmm. That could be challenging, although it seems intuitive that you should be able to prove the result from the one assumption. I must admit that I don't know off-hand how to do it, but maybe some ramblings may stimulate some thinking for you.
So the usual definition of matrix multiplication for square matrices is as follows. Let be square, matrices. The product matrix is the matrix consisting of entries as follows:
Let us assume that We wish to show that
Now, define the Kronecker delta symbol as follows:
It is standard to show that
Thus, by assumption, we have that
Now, we know that This you can prove using the definition of matrix multiplication. How would that look in summation notation?
Then, consider the fact that we can view the matrices as endomorphisms, for simplicities sake from to itself, then note that the existence of a left inverse for implies that is injective, thus is -dimensional, and so from an elementary theorem it follows that and so is surjective, thus bijective and so is invertible. Similarly, since possesses a right inverse we know that is surjective and it's easy to show that this is impossible if is not injective and thus is also invertible.
The rest of my proof stands.